[myren]

I don´t think this is an alien that applied downwards Löwenheim-Skolem to the reals. Rather I think this is an alien which applied upwards Löwenheim-Skolem to peano arithmetic. I think this because the alien talks about infinite natural numbers, aka non-standard natural numbers. It has used upwards Löwenheim-Skolem to get a model of peano arithmetic that has the cardinality of the continuum, and then there is of course a bijection between our reals and the thing that it calls the natural numbers. I see no sign that it´s using a non-standard model of the reals, but lots of signs that it´s using a non-standard model of the naturals.

[kmill]

The power of modern logic is that we can make predictions like this!

I wasn't sure whether the Löwenheim–Skolem alien was just translating things into our set theory for our sake, but I think your proposal is more likely.